Ehh…

So, it’s more a case that the system cannot prove it’s own consistency (an system cannot prove it won’t lead to a contradiction). So the proof is valid within the system, but the validity of the system is what was considered suspect (i.e. we cannot prove it won’t produce a contradiction from that system alone).

These days we use relative consistency proofs - that is we assume system A is consistent and model system B in it thus giving “If A is consistent, then so too must B”.

As much as I hate to admit it, classical set theory has been fairly robust - though intuitionistic logic makes better philosophical sense.